Sequences and Series - CBSE Class 11 Mathematics Notes

Chapter 8

SEQUENCES AND SERIES

Natural numbers are the product of human spirit. – DEDEKIND

8.1 Introduction

The term "sequence" in mathematics largely mirrors its usage in everyday language. To describe a set of elements as a sequence implies an inherent order, where each component can be distinctly identified as the first, second, third, and so forth. Illustrative instances include the population dynamics of organisms, such as humans or bacteria, observed at various temporal points; the cumulative sums of money accrued in a financial institution over successive years; or the diminishing valuations of specific assets. The utility of sequences extends across numerous domains of human endeavor.

Fibonacci (1175-1250)

Those sequences that exhibit discernible structural regularities are designated as progressions. Our prior academic work included an examination of arithmetic progression (A.P.). Within the scope of this present chapter, our discourse will extend beyond further exploration of A.P. to encompass the arithmetic mean, the geometric mean, the fundamental relationship between A.M. and G.M., and specific series formulations. These special series will include the summation of the first $n$ natural numbers, the summation of the squares of the first $n$ natural numbers, and the summation of the cubes of the first $n$ natural numbers.

8.2 Sequences

To illustrate, let's examine these scenarios:

Suppose a generation span of 30 years is assumed. We are tasked with determining the quantity of ancestors—such as parents, grandparents, and great-grandparents—a single individual would possess over a period of 300 years.

In this context, the cumulative number of generations is calculated as: $= \frac{300}{30} = 10$

The ancestral count for an individual across the first, second, third, up to the tenth generations is represented by the series 2, 4, 8, 16, 32, ..., 1024. This collection of numbers constitutes what is known as a sequence.

Now, consider the progressive quotients derived from dividing 10 by 3 through successive stages of calculation. This procedure yields the values 3, 3.3, 3.33, 3.333, and so forth. These resulting quotients similarly constitute a sequence. The individual numerical elements present within a sequence are referred to as its terms. Conventionally, we represent the terms of a sequence as $a_1, a_2, a_3, \ldots, a_n, \ldots$, where the subscripts indicate the respective position of each term. The element situated at the $n^{\text{th}}$ position within the sequence is designated as the $n^{\text{th}}$ term, symbolized by $a_n$. This $n^{\text{th}}$ term is also frequently called the general term of the sequence.

Therefore, the terms comprising the sequence depicting an individual's ancestors, as previously discussed, are:

$ a_1 = 2, a_2 = 4, a_3 = 8, \ldots, a_{10} = 1024. $

In a similar fashion, for the illustration involving successive quotients:

$ a_1 = 3, a_2 = 3.3, a_3 = 3.33, \ldots, a_6 = 3.33333, \text{ etc.} $

When a sequence comprises a determinate number of terms, it is designated as a finite sequence. For instance, the sequence representing ancestors is finite, given that it encompasses precisely 10 terms.

Conversely, a sequence is classified as infinite if it does not possess a finite number of terms. As an illustration, the sequence of successive quotients referenced earlier exemplifies an infinite sequence, characterized by its unending nature.

Frequently, the underlying principle that generates the distinct terms of a sequence can be articulated as an algebraic formula. For example, let's examine the sequence of even natural numbers: 2, 4, 6, ...

In this instance:

$ \begin{array}{l} a_1 = 2 = 2 \times 1 \quad a_2 = 4 = 2 \times 2 \ a_3 = 6 = 2 \times 3 \quad a_4 = 8 = 2 \times 4 \ \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \ \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \ a_{23} = 46 = 2 \times 23, \quad a_{24} = 48 = 2 \times 24, \text{ and so on.} \ \end{array} $

Indeed, it is evident that the $n^{\text{th}}$ term of this particular sequence can be represented by the expression $a_n = 2n$, where $n$ signifies a natural number. Analogously, for the sequence of odd natural numbers 1, 3, 5, ..., the $n^{\text{th}}$ term is specified by the formula $a_n = 2n - 1$, with $n$ again being a natural number.

Occasionally, a numerical arrangement like 1, 1, 2, 3, 5, 8, ... may not exhibit an immediately apparent explicit pattern, yet the sequence is produced through a recurrence relation defined as:

$ \begin{array}{l} a_1 = a_2 = 1 \ a_3 = a_1 + a_2 \ a_n = a_{n-2} + a_{n-1}, \quad n > 2 \ \end{array} $

This specific sequence is known as the Fibonacci sequence.

Within the sequence of prime numbers, namely 2, 3, 5, 7, ..., it is observed that no algebraic formula exists to directly compute the $n^{\text{th}}$ prime. Such sequences are characteristically defined solely through a verbal account or description.

It is not always the case that the terms of a sequence are defined by an explicit formula. Nevertheless, a conceptual framework or a generative rule is typically anticipated for producing the terms $a_1, a_2, a_3, \ldots, a_n, \ldots$ sequentially.

Consequently, a sequence may be conceptualized as a function for which the domain comprises the set of natural numbers or a designated subset thereof. Occasionally, the functional notation $a(n)$ is employed interchangeably with $a_n$.

8.3 Series

Given a sequence denoted by $a_1, a_2, a_3, \ldots, a_n$, the following

expression

$ a_1 + a_2 + a_3 + \ldots + a_n + \ldots $

constitutes the series corresponding to the specified sequence. A series is classified as either finite or infinite based on whether its originating sequence is finite or infinite. Frequently, series are expressed concisely using sigma notation, which employs the Greek letter $\Sigma$ (sigma) to signify the operation of summation. Consequently, the series $a_1 + a_2 + a_3 + \ldots + a_n$ is conventionally abbreviated as $\sum_{k=1}^{n} a_k$.

Remark The term "series," in its strict usage, denotes the indicated sum rather than the numerical result of the summation. For instance, $1 + 3 + 5 + 7$ represents a finite series comprising four terms. Conversely, when the expression "sum of a series" is employed, it refers to the scalar value obtained by aggregating its constituent terms; in the aforementioned example, the sum of the series is 16.

Let us now examine several illustrative examples.

Example 1 Write the first three terms in each of the following sequences defined by the following:

(i) $a_n = 2n + 5,$

(ii) $a_n = \frac{n - 3}{4}.$

Solution (i) Here $a_n = 2n + 5$

By substituting $n = 1, 2, 3$, we obtain:

$ a_1 = 2(1) + 5 = 7, a_2 = 9, a_3 = 11 $

Thus, the desired terms are 7, 9, and 11.

(ii) Here $a_n = \frac{n - 3}{4}$. Thus, $a_1 = \frac{1 - 3}{4} = -\frac{1}{2}$, $a_2 = -\frac{1}{4}$, $a_3 = 0$

Consequently, the initial three terms are $-\frac{1}{2}$, $-\frac{1}{4}$, and 0.

Example 2 What is the $20^{\text{th}}$ term of the sequence defined by

$ a_n = (n - 1)(2 - n)(3 + n)? $

Solution By substituting $n = 20$, we derive:

$ \begin{array}{l} a_{20} = (20 - 1)(2 - 20)(3 + 20) \ = 19 \times (-18) \times (23) = -7866. \end{array} $

Example 3 Let the sequence $a_n$ be defined as follows:

$ a_1 = 1, \quad a_n = a_{n-1} + 2 \text{ for } n \geq 2. $

Determine the initial five terms and formulate the corresponding series.

Solution We proceed as follows:

$ \begin{array}{l} a_1 = 1, \quad a_2 = a_1 + 2 = 1 + 2 = 3, \quad a_3 = a_2 + 2 = 3 + 2 = 5, \ a_4 = a_3 + 2 = 5 + 2 = 7, \quad a_5 = a_4 + 2 = 7 + 2 = 9. \end{array} $

Consequently, the first five terms of this sequence are 1, 3, 5, 7, and 9. The associated series is $1 + 3 + 5 + 7 + 9 + \ldots$.

EXERCISE 8.1

For the sequences defined in problems 1 through 6, determine the initial five terms, given their general expressions for the $n^{\text{th}}$ term:

1. $a_n = n(n + 2)$
2. $a_n = \frac{n}{n + 1}$
3. $a_n = 2^n$
4. $a_n = \frac{2n - 3}{6}$
5. $a_n = (-1)^{n-1} \cdot 5^{n+1}$
6. $a_n = n \cdot \frac{n^2 + 5}{4}$

From the sequences described in exercises 7 through 10, identify the specified terms using their given $n^{\text{th}}$ term formulas:

7. $a_n = 4n - 3; a_{17}, a_{24}$
8. $a_n = \frac{n^2}{2^n}; a_7$
9. $a_n = (-1)^{n-1} n^3; a_9$
10. $a_n = \frac{n(n - 2)}{n + 3}; a_{20}$

For each sequence presented in problems 11 through 13, determine the initial five terms and subsequently derive the associated series representation:

11. $a_1 = 3, a_n = 3a_{n-1} + 2$ for all $n > 1$

12. $a_1 = -1, a_n = \frac{a_{n-1}}{n}, n \geq 2$

13. $a_1 = a_2 = 2, a_n = a_{n-1} - 1, n > 2$

14. The Fibonacci sequence is formally defined as follows:

$ 1 = a _ {1} = a _ {2} \text{ and } a _ {n} = a _ {n - 1} + a _ {n - 2}, n > 2. $

Determine the ratio $\frac{a_{n + 1}}{a_n}$ for the specific values of $n = 1, 2, 3, 4, 5$.

8.4 Geometric Progression (G. P.)

Examine the subsequent numerical sequences:

(i) 2,4,8,16,..., (ii) $\frac{1}{9}, \frac{-1}{27}, \frac{1}{81}, \frac{-1}{243}$ ... (iii) .01,.0001,.000001,...

How do the elements within each of these sequences develop? It is observable that, excluding the initial term, every subsequent term advances according to a distinct pattern.

For sequence (i), we identify the first term as $a_1 = 2$, and the ratios between consecutive terms are consistently found to be $\frac{a_2}{a_1} = 2, \frac{a_3}{a_2} = 2, \frac{a_4}{a_3} = 2$, and so forth.

Likewise, in sequence (ii), we note that $a_1 = \frac{1}{9}$, and the ratios of successive terms are $\frac{a_2}{a_1} = -\frac{1}{3}, \frac{a_3}{a_2} = -\frac{1}{3}, \frac{a_4}{a_3} = -\frac{1}{3}$ and so on.

Analogously, how do the terms in sequence (iii) advance? It becomes evident that in each instance, every term, with the exception of the initial one, maintains a fixed ratio with respect to its directly preceding term. For sequence (i), this invariant ratio is 2; for sequence (ii), it is $-\frac{1}{3}$; and for sequence (iii), the constant ratio is 0.01. Sequences exhibiting this characteristic are termed geometric sequences or, more concisely, geometric progressions (GP).

A sequence denoted by $a_1, a_2, a_3, \ldots, a_n, \ldots$ is defined as a geometric progression if all its terms are non-zero and the ratio $\frac{a_{k+1}}{a_k}$ remains a constant value, $r$, for all $k \geq 1$.

Designating the initial term as $a_1 = a$, we derive a geometric progression in the form $a, ar, ar^2, ar^3, \ldots$. Here, $a$ represents the first term, and $r$ is designated as the common ratio of the G.P. The common ratios for the geometric progressions (i), (ii), and (iii) previously mentioned are 2, $-\frac{1}{3}$, and 0.01, respectively.

Similar to arithmetic progressions, determining the $n^{\text{th}}$ term or the cumulative sum of $n$ terms for a geometric progression comprising numerous elements would present considerable difficulty without recourse to the formulas to be established in the subsequent section. For these formulas, we will employ the following standard notations:

$a =$ the initial term, $r =$ the constant ratio, $l =$ the terminal term,

$n =$ the quantity of terms,

$\mathbf{S}_n =$ the sum of the initial $n$ terms.

8.4.1 General term of a G.P.

Let us analyze a G.P. characterized by an initial non-zero term $a$ and a common ratio $r$. Let's enumerate some of its terms. The second term is derived by multiplying $a$ by $r$, yielding $a_2 = ar$. Likewise, the third term is found by multiplying $a_2$ by $r$, resulting in $a_3 = a_2r = ar^2$, and this pattern continues.

Presented below are these terms and several additional ones.

$1^{\mathrm{st}}$ term $= a_{1} = a = ar^{1 - 1}$, $2^{\mathrm{nd}}$ term $= a_{2} = ar = ar^{2 - 1}$, $3^{\mathrm{rd}}$ term $= a_{3} = ar^{2} = ar^{3 - 1}$

$4^{\text{th}}$ term $= a_{4} = ar^{3} = ar^{4 - 1}$, $5^{\text{th}}$ term $= a_{5} = ar^{4} = ar^{5 - 1}$

Is a discernible pattern evident? What would be the $16^{\text{th}}$ term in this sequence?

$ a _ {1 6} = a r ^ {1 6 - 1} = a r ^ {1 5} $

Consequently, the observed pattern indicates that the $n^{\text{th}}$ term of a G.P. can be expressed as $a_{n} = ar^{n - 1}$.

Hence, a G.P. can be formulated as $a, ar, ar^2, ar^3, \ldots, ar^{n-1}$ if it is finite, or as $a, ar, ar^2, ar^3, \ldots$ if it is infinite.

The sums $a + ar + ar^2 + \ldots + ar^{n-1}$ and $a + ar + ar^2 + \ldots$ are termed finite or infinite geometric series, correspondingly.

8.4.2 Sum to $n$ terms of a G.P.

Consider a Geometric Progression with its initial term designated as $a$ and its common ratio as $r$. The summation of its initial $n$ terms is represented by $S_n$. Consequently,

$ \mathrm {S} _ {n} = a + a r + a r ^ {2} + \dots + a r ^ {n - 1} \tag {1} $

Case 1: Should the common ratio $r$ be equal to 1, then $\mathbf{S}_n$ becomes the sum of $n$ instances of $a$, resulting in $\mathbf{S}_n = na$.

Case 2: If $r$ is not equal to 1, multiplying equation (1) by $r$ yields:

$ r \mathrm {S} _ {n} = a r + a r ^ {2} + a r ^ {3} + \dots + a r ^ {n} \tag {2} $

Upon subtracting equation (2) from equation (1), the result is $(1 - r)\mathbf{S}_n = a - ar^n = a(1 - r^n)$.

This leads to

$ \mathrm {o r} \quad \mathrm {S} _ {n} = \frac {a (r ^ {n} - 1)}{r - 1} $

Example 4 Find the $10^{\text{th}}$ and $n^{\text{th}}$ terms of the G.P. 5, 25, 125, ...

Solution: In this sequence, the initial term $a$ is 5, and the common ratio $r$ is also 5. Consequently, the tenth term, $a_{10}$, is calculated as $5(5)^{10 - 1} = 5(5)^{9} = 5^{10}$, and the general $n^{\text{th}}$ term, $a_{n}$, is $ar^{n - 1} = 5(5)^{n - 1} = 5^{n}$.

Example 5 Which term of the G.P., 2,8,32, ... up to $n$ terms is 131072?

Solution: Assume that 131072 represents the $n^{\text{th}}$ term of the specified geometric progression. For this series, the first term $a$ is 2, and the common ratio $r$ is 4.

Thus, we have $131072 = a_{n} = 2(4)^{n - 1}$, which simplifies to $65536 = 4^{n - 1}$.

This equality implies $4^{8} = 4^{n - 1}$.

Consequently, $n - 1 = 8$, which means $n = 9$. Therefore, 131072 is identified as the ninth term of the geometric progression.

Example 6 In a G.P., the $3^{\mathrm{rd}}$ term is 24 and the $6^{\mathrm{th}}$ term is 192. Find the $10^{\mathrm{th}}$ term.

Solution: We are provided that $a_3 = ar^2 = 24$ ... (1)

and $a_6 = ar^5 = 192$ ... (2)

By dividing Equation (2) by Equation (1), we ascertain that $r = 2$. Subsequently, substituting $r = 2$ into Equation (1) yields $a = 6$.

Therefore, the tenth term $a_{10}$ is $6(2)^{9} = 3072$.

Example 7 Find the sum of first $n$ terms and the sum of first 5 terms of the geometric

series $1 + \frac{2}{3} + \frac{4}{9} + \dots$

Solution: In this series, the first term $a$ is 1, and the common ratio $r$ is $\frac{2}{3}$. Consequently,

$ S_{n} = \frac{a (1 - r^{n})}{1 - r} = \frac{\left[1 - \left(\frac{2}{3}\right)^{n}\right]}{1 - \frac{2}{3}} = 3 \left[1 - \left(\frac{2}{3}\right)^{n}\right] $

Specifically, for the sum of the first five terms, $S_{5} = 3\left[1 - \left(\frac{2}{3}\right)^{5}\right] = 3 \times \frac{211}{243} = \frac{211}{81}$.

Example 8 How many terms of the G.P. $3, \frac{3}{2}, \frac{3}{4}, \ldots$ are needed to give the

sum $\frac{3069}{512}$

Solution: Let $n$ denote the required quantity of terms. We are provided that the first term $a = 3$, the common ratio $r = \frac{1}{2}$, and the sum of $n$ terms $S_{n} = \frac{3069}{512}$.

Given the formula for the sum of $n$ terms, $S_{n} = \frac{a(1 - r^{n})}{1 - r}$,

Therefore

$ \frac{3069}{512} = \frac{3\left(1 - \frac{1}{2^n}\right)}{1 - \frac{1}{2}} = 6\left(1 - \frac{1}{2^n}\right) $

or

$ \frac{3069}{3072} = 1 - \frac{1}{2^n} $

or

$ \frac{1}{2^n} = 1 - \frac{3069}{3072} = \frac{3}{3072} = \frac{1}{1024} $

or

$ 2^n = 1024 = 2^{10}, \text{ which gives } n = 10. $

Example 9 The sum of first three terms of a G.P. is $\frac{13}{12}$ and their product is $-1$. Find the common ratio and the terms.

Solution: Designate the first three terms of the geometric progression as $\frac{a}{r}$, $a$, and $ar$. In this scenario,

$ \frac{a}{r} + ar + a = \frac{13}{12} \tag{1} $

and

$ \left(\frac{a}{r}\right)(a)(ar) = -1 \tag{2} $

Deducing from equation (2), we determine that $a^3 = -1$, which implies $a = -1$ when restricting consideration to real roots.

Upon substituting this value of $a = -1$ into equation (1), the expression transforms into:

$ -\frac{1}{r} - 1 - r = \frac{13}{12} \text{ or } 12r^2 + 25r + 12 = 0. $

This equation represents a quadratic relationship in $r$. Solving it yields two distinct values for $r$: specifically, $r = -\frac{3}{4}$ or $r = -\frac{4}{3}$.

Consequently, the three terms of the geometric progression are identified as: $\frac{4}{3}, -1, \frac{3}{4}$ when $r = -\frac{3}{4}$, and $\frac{3}{4}, -1, \frac{4}{3}$ when $r = -\frac{4}{3}$.

Example 10 Determine the sum of the sequence 7, 77, 777, 7777, ... up to $n$ terms.

Solution This sequence does not constitute a geometric progression (G.P.) directly. However, it can be transformed to reveal a relationship with a G.P. by expressing its terms as follows:

$ S_n = 7 + 77 + 777 + 7777 + \dots \text{ to } n \text{ terms}

$

$ \begin{array}{l} = \frac{7}{9} [9 + 99 + 999 + 9999 + \dots \text{to } n \text{ term}] \ = \frac{7}{9} [(10 - 1) + (10^2 - 1) + (10^3 - 1) + (10^4 - 1) + \dots n \text{ terms}] \ = \frac{7}{9} [(10 + 10^2 + 10^3 + \dots n \text{ terms}) - (1 + 1 + 1 + \dots n \text{ terms})] \ = \frac{7}{9} \left[ \frac{10(10^n - 1)}{10 - 1} - n \right] = \frac{7}{9} \left[ \frac{10(10^n - 1)}{9} - n \right]. \end{array} $

Example 11 Consider an individual who has 2 parents, 4 grandparents, 8 great-grandparents, and this pattern continues. Determine the total count of their ancestors across the ten generations immediately preceding their own.

Solution In this scenario, we identify the initial term as $a = 2$, the common ratio as $r = 2$, and the number of terms (generations) as $n = 10$.

Employing the sum formula for a geometric series, $\mathrm{S}_n = \frac{a(r^n - 1)}{r - 1}$, we proceed with the calculation:

We find that $\mathrm{S}_{10} = 2(2^{10} - 1) = 2046$.

Therefore, the cumulative number of ancestors for the person over the ten preceding generations amounts to 2046.

8.4.3 Geometric Mean (G.M.)

For any two positive numbers $a$ and $b$, their geometric mean is defined as the number $\sqrt{ab}$. As an illustration, the geometric mean of 2 and 8 is 4. It can be noted that the sequence 2, 4, 8 constitutes consecutive terms of a geometric progression (G.P.). This observation serves as a foundation for extending the notion of geometric means between two numbers.

Between any two positive numbers, $a$ and $b$, an arbitrary quantity of numbers can be interpolated such that the entire resulting sequence forms a geometric progression.

Consider $n$ numbers, denoted $\mathrm{G}_1, \mathrm{G}_2, \ldots, \mathrm{G}_n$, situated between two positive numbers $a$ and $b$. These numbers are arranged such that the sequence $a, \mathrm{G}_1, \mathrm{G}_2, \mathrm{G}_3, \ldots, \mathrm{G}_n, b$ forms a geometric progression. Consequently, $b$ functions as the $(n+2)^{\text{th}}$ term of this progression, leading to the following relationship:

$ b = a r^{n + 1}, \quad \text{or} \quad r = \left(\frac{b}{a}\right)^{\frac{1}{n + 1}}. $

From this, the individual geometric means can be expressed as: $\mathrm{G}_1 = ar = a\left(\frac{b}{a}\right)^{\frac{1}{n + 1}}$, $\mathrm{G}_2 = ar^2 = a\left(\frac{b}{a}\right)^{\frac{2}{n + 1}}$, $\mathrm{G}_3 = ar^3 = a\left(\frac{b}{a}\right)^{\frac{3}{n + 1}}$,

$ \mathrm{G}_n = a r^n = a \left(\frac{b}{a}\right)^{\frac{n}{n + 1}} $

Example 12 Determine three numbers to be inserted between 1 and 256 such that the resulting sequence constitutes a geometric progression.

Solution. Let $G_1, G_2, G_3$ represent the three numbers positioned between 1 and 256, forming a G.P. in the sequence $1, G_1, G_2, G_3, 256$.

Consequently, $256 = r^4$, which yields $r = \pm 4$ (considering only real roots).

If $r = 4$, the terms are computed as $G_1 = ar = 4$, $G_2 = ar^2 = 16$, and $G_3 = ar^3 = 64$.

Analogously, when $r = -4$, the corresponding numbers are $-4, 16$, and $-64$.

Therefore, the numbers 4, 16, 64 can be inserted between 1 and 256 to form geometric progressions.

8.5 Relationship Between A.M. and G.M.

Let $A$ represent the arithmetic mean (A.M.) and $G$ denote the geometric mean (G.M.) for any two given positive real numbers, $a$ and $b$. These are defined as follows:

$ A = \frac{a + b}{2} \quad \text{and} \quad G = \sqrt{ab} $

To establish the relationship between these means, we consider their difference:

$ \begin{array}{l} A - G = \frac{a + b}{2} - \sqrt{ab} = \frac{a + b - 2\sqrt{ab}}{2} \ = \frac{\left(\sqrt{a} - \sqrt{b}\right)^2}{2} \geq 0 \tag{1} \end{array} $

From the expression (1), it is evident that $A - G$ is always greater than or equal to zero, which implies the fundamental inequality $A \geq G$.

Example 13 Suppose the arithmetic mean and geometric mean of two positive numbers, $a$ and $b$, are 10 and 8, respectively. Determine the values of these numbers.

Solution We are provided with the following information: The arithmetic mean: $\mathrm{A.M.} = \frac{a + b}{2} = 10$ ... (1) The geometric mean: $\mathrm{G.M.} = \sqrt{ab} = 8$ ... (2)

From equation (1), we can deduce the sum of the numbers:

$ a + b = 20 \tag{3} $

Squaring both sides of equation (2) yields the product of the numbers:

$ ab = 64 \tag{4} $

Utilizing the algebraic identity $(a - b)^2 = (a + b)^2 - 4ab$, and substituting the values from (3) and (4), we calculate:

$ (a - b)^2 = (20)^2 - 4(64) = 400 - 256 = 144 $

Taking the square root of both sides, we find:

$ a - b = \pm 12 \tag{5}

$

Now, we solve the system of linear equations formed by (3) and (5).

Case 1: $a - b = 12$ Adding $a+b=20$ and $a-b=12$: $2a = 32 \implies a = 16$ Substituting $a=16$ into $a+b=20$: $16+b=20 \implies b=4$

Case 2: $a - b = -12$ Adding $a+b=20$ and $a-b=-12$: $2a = 8 \implies a = 4$ Substituting $a=4$ into $a+b=20$: $4+b=20 \implies b=16$

Therefore, the two numbers are either 4 and 16, or 16 and 4.

EXERCISE 8.2

1. Determine the $20^{\text{th}}$ term and the general $n^{\text{th}}$ term for the given geometric progression: $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$.

2. For a geometric progression where the $8^{\text{th}}$ term equals 192 and the common ratio is 2, ascertain its $12^{\text{th}}$ term.

3. Given that $p$, $q$, and $s$ represent the $5^{\text{th}}$, $8^{\text{th}}$, and $11^{\text{th}}$ terms of a geometric progression, respectively, demonstrate that $q^2 = ps$.

4. In a geometric progression, the fourth term is equivalent to the square of its second term, and the initial term is $-3$. Calculate its $7^{\text{th}}$ term.

5. Identify the term number for the specified value in each of the following sequences:

   (a) For the sequence $2,2\sqrt{2},4,\ldots$, which term corresponds to the value 128?

   (b) For the sequence $\sqrt{3}, 3, 3\sqrt{3}, \ldots$, which term corresponds to the value 729?

   (c) For the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots$, which term corresponds to the value $\frac{1}{19683}$?

6. Determine the values of $x$ such that the sequence of numbers $-\frac{2}{7}, x, -\frac{7}{2}$ forms a geometric progression.

For each of the geometric progressions presented in Exercises 7 through 10, calculate the sum up to the specified number of terms:

7. The series $0.15, 0.015, 0.0015, \ldots$ for 20 terms.

8. The series $\sqrt{7}, \sqrt{21}, 3\sqrt{7}, \ldots$ for $n$ terms.

9. The series $1, -a, a^2, -a^3, \ldots$ for $n$ terms (assuming $a \neq -1$).

10. The series $x^3, x^5, x^7, \ldots$ for $n$ terms (assuming $x \neq \pm 1$).

11. Compute the value of the summation $\sum_{k=1}^{11} (2 + 3^k)$.

12. A geometric progression has a sum of its initial three terms equal to $\frac{39}{10}$, and their product is 1. Determine both the common ratio and these three terms.

13. For the geometric progression $3, 3^2, 3^3, \ldots$, ascertain the number of terms required to yield a sum of 120.

14. In a geometric progression, the sum of the first three terms is 16, and the sum of the subsequent three terms (i.e., the fourth, fifth, and sixth terms) is 128. Calculate the first term, the common ratio, and the sum of the first $n$ terms of this progression.

15. For a geometric progression where the first term $a = 729$ and the $7^{\text{th}}$ term is 64, compute $S_7$.

16. Identify a geometric progression such that the sum of its initial two terms is $-4$, and its fifth term is four times its third term.

17. Assuming that $x, y,$ and $z$ denote the $4^{\text{th}}$, $10^{\text{th}}$, and $16^{\text{th}}$ terms of a geometric progression, respectively, establish that $x, y, z$ also constitute a geometric progression.

18. Determine the sum of the first $n$ terms for the sequence $8, 88, 888, 8888, \ldots$.

19. Calculate the sum obtained by multiplying the corresponding terms of the two sequences: $2, 4, 8, 16, 32$ and $128, 32, 8, 2, \frac{1}{2}$.

20. Demonstrate that a new sequence formed by the product of the corresponding terms from two geometric progressions, $a, ar, ar^2, \ldots, ar^{n-1}$ and $A, AR, AR^2, \ldots, AR^{n-1}$, itself constitutes a geometric progression, and subsequently identify its common ratio.

21. Determine four numbers that constitute a geometric progression, given that the third term exceeds the first term by 9, and the second term exceeds the fourth term by 18.

22. Given that $a, b,$ and $c$ are the $p^{\text{th}}$, $q^{\text{th}}$, and $r^{\text{th}}$ terms of a geometric progression, respectively, establish the following identity:

    $ a^{q-r} b^{r-p} c^{p-q} = 1. $

23. Let $a$ and $b$ denote the first and $n^{\text{th}}$ terms of a geometric progression, respectively, and let $P$ be the product of its first $n$ terms. Demonstrate that $P^2 = (ab)^n$.

24. Prove that the ratio between the sum of the initial $n$ terms of a geometric progression and the sum of terms ranging from the $(n+1)^{\text{th}}$ term to the $(2n)^{\text{th}}$ term is equal to $\frac{1}{r^n}$.

25. Given that $a, b, c,$ and $d$ form a geometric progression, establish the following relationship:

    $ (a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2. $

26. Identify two numbers that, when positioned between 3 and 81, establish a geometric progression.

27. Determine the value of $n$ for which the expression $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$ serves as the geometric mean between $a$ and $b$.

28. Given that the sum of two numbers is six times their geometric mean, establish that their ratio is $(3 + 2\sqrt{2}) : (3 - 2\sqrt{2})$.

29. Let A and G denote the arithmetic mean and geometric mean, respectively, of two positive numbers. Prove that these numbers can be determined by the expression $A \pm \sqrt{(A + G)(A - G)}$.

30. In a specific bacterial culture, the population doubles every hour. If the initial count was 30 bacteria, determine the number of bacteria present at the end of the $2^{\text{nd}}$ hour, the $4^{\text{th}}$ hour, and the $n^{\text{th}}$ hour.

31. Calculate the total sum that Rs 500 will accumulate to in 10 years, given it is deposited in a bank offering an annual interest rate of 10% compounded yearly.

32. Given that the arithmetic mean and geometric mean of the roots of a quadratic equation are 8 and 5, respectively, construct the quadratic equation.

Miscellaneous Examples

Example 14 Given that $a, b, c, d,$ and $p$ are distinct real numbers satisfying the inequality $(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0$, demonstrate that $a, b, c,$ and $d$ constitute a geometric progression (G.P.).

Solution Given that

$ (a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0 \tag{1} $

However, the left-hand side (L.H.S.) can be expanded as follows:

$ = (a^2p^2 - 2abp + b^2) + (b^2p^2 - 2bcp + c^2) + (c^2p^2 - 2cdp + d^2), $

This expression simplifies to $(ap - b)^2 + (bp - c)^2 + (cp - d)^2 \geq 0$ ... (2)

Given that the sum of squared real numbers cannot be negative, and considering the initial inequality (1) alongside the derived expression (2), it logically follows that $(ap - b)^2 + (bp - c)^2 + (cp - d)^2 = 0$.

This condition necessitates that each squared term must individually be zero; thus, $ap - b = 0, bp - c = 0,$ and $cp - d = 0$.

Consequently, we deduce that $\frac{b}{a} = \frac{c}{b} = \frac{d}{c} = p$.

Therefore, $a, b, c,$ and $d$ form a geometric progression.

Miscellaneous Exercise on Chapter 8

1. Suppose a function $f$ satisfies the property $f(x + y) = f(x)f(y)$ for all natural numbers $x, y \in \mathbf{N}$. Given that $f(1) = 3$ and the summation $\sum_{x=1}^{n} f(x) = 120$, determine the value of $n$.

2. For a geometric progression (G.P.), the sum of its initial terms is 315. The first term is 5, and the common ratio is 2. Calculate the last term and the total number of terms in this progression.

3. A geometric progression (G.P.) has a first term of 1. The sum of its third and fifth terms is 90. Determine the common ratio of this G.P.

4. Three numbers in a geometric progression (G.P.) have a sum of 56. If 1, 7, and 21 are successively subtracted from these numbers, an arithmetic progression is formed. Identify these three numbers.

5. Consider a geometric progression (G.P.) comprising an even quantity of terms. If the aggregate sum of all terms is five times the sum of the terms positioned at odd indices, determine its common ratio.

6. Given the equality $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$, where $x \neq 0$, demonstrate that $a, b, c,$ and $d$ constitute a geometric progression (G.P.).

7. For a geometric progression (G.P.) of $n$ terms, let $S$ represent the sum of these terms, $P$ their product, and $R$ the sum of their reciprocals. Establish the identity $P^2 R^n = S^n$.

8. If $a, b, c,$ and $d$ form a geometric progression (G.P.), prove that the terms $(a^n + b^n), (b^n + c^n),$ and $(c^n + d^n)$ also form a geometric progression.

9. Let $a$ and $b$ be the roots of the quadratic equation $x^{2} - 3x + p = 0$, and let $c$ and $d$ be the roots of $x^{2} - 12x + q = 0$. Given that $a, b, c, d$ constitute a geometric progression (G.P.), demonstrate that the ratio $(q + p) : (q - p)$ is equal to $17:15$.

10. For two positive numbers, $a$ and $b$, the ratio of their arithmetic mean (A.M.) to their geometric mean (G.M.) is $m : n$. Establish that the ratio $a : b$ is given by $\left(m + \sqrt{m^2 - n^2}\right) : \left(m - \sqrt{m^2 - n^2}\right)$.

11. Determine the sum of each of the following series for $n$ terms:

(i) $5 + 55 + 555 + \dots$

(ii) $.6 + .66 + .666 + \dots$

12. Identify the $20^{\text{th}}$ term in the sequence defined by $2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots + n$ terms.
13. A farmer acquires a pre-owned tractor for Rs 12000. An initial payment of Rs 6000 is made, with the remaining sum to be settled through annual instalments of Rs 500, augmented by a $12%$ interest charge on the outstanding balance. What will be the total expenditure incurred by the farmer for the tractor?
14. Shamshad Ali purchases a scooter priced at Rs 22000. He tenders Rs 4000 as a down payment and undertakes to remit the remainder via annual instalments of Rs 1000, along with a $10%$ interest rate applied to the diminishing unpaid principal. Calculate the total cost Shamshad Ali will pay for the scooter.
15. An individual dispatches a letter to four acquaintances. Each recipient is instructed to replicate the letter and forward it to four distinct new individuals, thereby perpetuating the chain. Assuming uninterrupted propagation of this chain and a postage cost of 50 paise per letter, determine the total amount expended on postage upon the mailing of the $8^{\text{th}}$ iteration of letters.
16. A sum of Rs 10000 is deposited by a person in a bank, accruing simple interest at an annual rate of $5%$. Ascertain the total amount available in the $15^{\text{th}}$ year following the initial deposit, and subsequently compute the aggregate amount after 20 years.
17. A manufacturer estimates that a machine, initially procured for Rs 15625, will undergo an annual depreciation of $20%$ of its value. Determine the projected worth of this machine at the conclusion of 5 years.
18. A task was initially assigned to 150 workers with the expectation of completion within a specific timeframe. However, 4 workers withdrew on the second day, another 4 on the third day, and this pattern of attrition continued. As a result, the work required an additional 8 days for completion. Calculate the total number of days taken to finish the task.

Summary

A sequence is understood as an ordered arrangement of numbers established according to a specific rule. Alternatively, a sequence can be defined as a function whose domain is either the set of natural numbers or a finite subset thereof, such as ${1, 2, 3, \dots, k}$. A sequence containing a finite count of terms is referred to as a finite sequence. Conversely, a sequence is designated as infinite if it does not possess a finite number of terms.

Given a sequence denoted by $a_1, a_2, a_3, \ldots$, the summation expressed as $a_1 + a_2 + a_3 + \ldots$ is termed a series. A series is classified as a finite series if it comprises a finite number of terms.

A sequence is identified as a geometric progression or G.P. if the ratio of any term to its immediate preceding term remains constant throughout. This invariant factor is known as the common ratio. Typically, the initial term of a G.P. is represented by $a$, and its common ratio by $r$. The general, or $n^{\text{th}}$, term of a G.P. is mathematically given by $a_n = a r^{n-1}$. The sum $S_n$ of the first $n$ terms of a G.P. is provided by:

$ S_n = \frac{a(r^n - 1)}{r - 1} \text{ or } \frac{a(1 - r^n)}{1 - r}, \text{ if } r \neq 1 $

For any two positive numbers, $a$ and $b$, their geometric mean (G.M.) is determined by $\sqrt{ab}$; this signifies that the sequence $a, G, b$ constitutes a G.P.

Historical Note

Historical records indicate that the Babylonians possessed knowledge of arithmetic and geometric progressions approximately four millennia ago. Boethius, writing in 510 CE, attests to the familiarity of early Greek scholars with these types of sequences. The Indian mathematician Aryabhatta (born 476 CE) is credited as the first to articulate the formulae for the summation of squares and cubes of natural numbers within his seminal treatise, the Aryabhatiyam, composed around 499 CE. Furthermore, he provided a method for calculating the sum of the first $n$ terms of an arithmetic sequence, commencing from its $p^{\text{th}}$ element. Prominent Indian mathematicians including Brahmgupta (

598), Mahavira (850), and Bhaskara (1114-1185) similarly engaged with the problem of summing squares and cubes. The Italian mathematician Leonardo Fibonacci (1170-1250) introduced another distinct class of sequence, known as the Fibonacci sequence, which holds significant mathematical utility. The seventeenth century marked a period of systematic categorization for various series forms. Notably, in 1671, James Gregory employed the nomenclature 'infinite series' when referring to an infinite sequence. The precise articulation of concepts pertaining to sequences and series became feasible only subsequent to the rigorous advancement of algebraic and set-theoretic methodologies.